Abstract

Delayed plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are presented and investigated. Firstly, we consider continuous cultural control strategy in which continuous replanting of healthy plants is taken. The existence and local stability of disease-free equilibrium and positive equilibrium are studied by analyzing the associated characteristic transcendental equation. And then, plant disease model with impulsive replanting of healthy plants is also considered; the sufficient condition under which the infected plant-free periodic solution is globally attritive is obtained. Moreover, permanence of the system is studied. Some numerical simulations are also given to illustrate our results.

1. Introduction

Plant viruses or pathogens are an important constraint to crop production worldwide and cause major production and economic losses in agriculture and forestry. For example, soybean rust (a fungal disease in soybeans) has caused a significant economic loss, and just by removing 20% of the infection, the farmers may benefit with an approximately 11 million-dollar profit [1]. Several plant diseases caused by plant viruses in cassava (Manihot esculenta) and sweet potato (Ipomoea batatas) are among the main constraints to sustainable production of these vegetatively propagated staple food crops in lesser-developed countries [2–4]. A strain of the virus causing cassava mosaic disease gives rise to losses in Africa [5]. Therefore, people have turned more attention to plant diseases. Several conferences have been held to discuss how to control or prevent plant virus. Therefore, farmers have been evolving practices for controlling plant diseases, which involves a number of dynamic processes such as the growth of plants and the spread of diseases. Recently, the integrated disease management (IDM) which combines biological, cultural, and chemical tactics and so on to reduce the numbers of infected individuals to a tolerable level and aims to minimize losses and maximize returns [6, 7] has been developed gradually. IDM includes four main control strategies for vegetatively propagated plant diseases, which are containing transmission vectors, improving the production of planting material, controlling the crop sanitation through removal of infected plants, and breeding plants for resistance to the virus. Breeding plants for resistance to the virus as an cultural strategy has been widely used into practice [8–11]. In the system of IDM, mathematical modeling has shown its unique value on describing, analyzing, and predicting plant epidemics [12–16]. Meng and Li have investigated vegetatively propagated plant diseases and developed a mathematical model with continuous control strategies and impulsive cultural control strategies [17], which leads to
where denote the number of susceptible and infected plants, respectively. is the transmission rate, denotes potentially density dependent, either denotes harvest time or the end of reproductive life time of plants, represents the total rate at which the susceptible plants enter the system, is the removal rate for the infected plants, is the recovery rate of the cured diseased plants, and the infected plants suffer an extra disease-related death with constant rate . In system (1.1), the authors refer to two-control strategy: one is continuous control and the other is impulsive control by implementing periodic replanting of healthy plants or removing infected plants at a critical time. A model for the spread of an infectious disease (involving only susceptibles and infective individuals) transmitted by a vector after an incubation time was proposed by Cooke [18]. This is called the phenomena of time delay. Many authors have directly incorporated time delays in modeling equations, and, as a result, the models take the form of delay differential equations [19–23]. Motivated by Meng, we get the following reasonable plant disease models by introducing time delay:

From the point of biology, we only consider system (1.2) and (1.3) in the biological meaning region: . Let
where is positive, bounded, and continuous function for . Motivated by the application of systems (1.2) and (1.3) to population dynamics (refer to [24]), we assume that solutions of systems (1.1) satisfy the following initial conditions:

In this section, we consider system (1.2) with continuous replanting and removing and without impulsive effect. By Smith [25, Theorem 5.2.1] or Zhao and Zou [26], for any , there is a unique solution of system (1.2) with , for any and , for all in its maximal interval of existence.

Define ; then we have
Let ; we have
Then
as . Hence, system (1.2) is uniformly bounded.

Since and denote the number of susceptible and infected plants, respectively, it is easy to observe that system (1.2) has a disease-free of the form , and a unique infection equilibrium provided that we have the following condition:
where

2.1. The Stability of the Disease-Free Equilibrium

We may firstly consider the stability of the disease-free equilibria . Let , ; then system (1.2) can be rewritten as the following equivalent system:
Thus, the disease-free equilibrium of system (1.2) is transformed into zero equilibrium of system (2.6). Linearizing system (2.6) about the equilibrium (0, 0) yields the following linear system:
with characteristic equation:
that is,

The stability of trivial solution of system (1.2) depends on the locations of roots of characteristic equation (2.9). When all roots of (2.9) locate in the left half-plane of complex plane, the trivial solution of system (1.2) is stable; otherwise, it is unstable. In the following, we will investigate the distribution of roots of (2.9). Obviously, . Let
For (2.10), the root of (2.10) with always has negative real part provided that .

In addition, is a root of (2.10) if and only if satisfies the following equation:

Separating the real and imaginary parts of (2.11) gives the following equations:

Thus we can have
Then if , (2.13) has not positive real root, which leads to (2.10) that has not purely imaginary root. By the Rouche Theory, we know that all the roots of (2.9) have always negative real parts. So the equilibrium of system (1.2) is locally asymptotically stable.

Define
For system (1.2), we have the following result on stability of the disease-free equilibrium .

Theorem 2.1. For system (1.2), the following statements are true.(i)If , then the disease-free equilibrium of system (1.2) is unstable.(ii)If , then the disease-free equilibrium of system (1.2) is locally asymptotically stable.

In this section, we show that the disease equilibrium is asymptotically stable in the case that time delay is less than the unity; then we have the following theorem.

Theorem 2.2. For system (1.2), if , then the positive equilibrium of system (1.2) is asymptotically stable.

Proof. Under the hypothesis , let , then system (1.2) can be rewritten as the following equivalent system:
Thus, the positive equilibrium of system (1.2) is transformed into zero equilibrium of system (2.15). Linearizing system (2.15) about the equilibrium (0, 0) yields the following linear system:
with characteristic equation:
The stability of trivial solution of system (1.2) depends on the locations of roots of characteristic equation (2.17). For the sake of simplicity, let
Then (2.17) can be briefly denoted as the following equation:
For (2.19), we can claim that the two roots of (2.19) have always negative real parts. We will prove it in the following two steps.Step 1. If , (2.19) can be simplified as
Note that
Therefore, the two roots of (2.19) with have always negative real parts.Step 2. If is a root of (2.15) if and only if satisfies the following equation:
Separating the real and imaginary parts of (2.22) gives the following equations:
One can obtain
We can easily see that
So (2.24) has not positive real root, which leads to (2.19) that has not purely imaginary root. By the Rouche Theory, we know that all the roots of (2.19) have always negative real parts. So the equilibrium of system (1.2) is asymptotically stable. The proof is complete.

3.1. Boundedness

Let the initial data be . Then, one can easily prove that the solutions of system (1.3) are positive for all . Now, let . We calculate the upper right derivative of along with a solution of system (1.3) with :
Since , one can deduce that
where . We consider the following impulse differential inequalities:
According to impulse differential inequalities theory, we get
as .

So is uniformly ultimately bounded. Hence, by the definition of , for any , there exists a constant such that and for each solution of (1.3) with being large enough.

3.2. Global Attractivity of the Disease-Free Periodic Solution of System (1.3)

In the following, we shall prove that the disease-free periodic is stable if it exists. For this purpose, we give firstly some basic properties of the following subsystem:
We can find a unique positive periodic solution , which is globally asymptotically stable by using stroboscopic map. As a consequence, system (3.5) always has a disease-free periodic solution . Now, we give the conditions which assure the global attractivity of disease-free periodic solution of the system (1.3).

Denote

Theorem 3.1. The disease-free periodic solution of system (1.3) is globally attractive provided that

Proof. Let be any solution of system (1.3). From the first equation of system (1.3), it follows that , for ; then we consider the following impulse differential system:
Obviously, system (3.8) has a globally asymptotically stable positive periodic solution:
By the comparison theorem in impulsive differential equation, for any sufficiently small positive , there exists an integer such that
Therefore, from the second equation of system (1.3), we have
Now we consider the following comparison equation:
Since , we have
We may choose three sufficiently small positive constants such that
According to the theory of delay differential equation [24], we obtain that . By impulsive comparison theorem, we have with being large enough. Therefore, we obtain that .Then for a sufficiently small and all being large enough, we have . Without loss of generality, we may assume as . From the first equation of system (1.3), we have
Consider the following comparison system:
Then, system (3.16) has a positive periodic solution:
which is globally asymptotically stable. Thus, for a sufficiently small , when is large enough, we have
From the first equation of system (1.3), we also have
Consider the following comparison system:
System (3.20) has a globally asymptotically stable positive periodic solution:
Thus, for a sufficiently small , when is large enough, we have
Combining (3.18) with (3.22), we obtain
Let ; we have , which implies . The proof is completed.

Definition 3.2. System (1.3) is said to be permanent if there exist constants (independent of initial value) and a finite time such that for every positive solution with initial conditions (1.3) satisfies for all . Here may depend on the initial condition.

Denote

Lemma 3.3. If , then there exists a positive constant such that

Proof. Define
Calculating the derivative of along with the solution of (1.3), we can get
for .Since , then . Note that and .Solving the aforementioned inequality, we can have that
We can choose being small enough such that
For any positive constant , we claim that the inequality cannot hold for all . Otherwise, there exists a positive constant such that for all . From the first equation of (1.3), we have
Similarly, we know that there exists such , for that
Then, by (3.30), we have that, for ,
Let
We show that for all . Otherwise, there exists a nonnegative constant such that for and . Thus from the second equation of (1.3) and (3.30), we easily see that
which is a contradiction. Hence we get that for all . Therefore, for all , we have
which implies as . This is a contradiction to for being large enough. Therefore, for any positive constant , the inequality cannot hold for all .On the one hand, if holds true for all being large enough, then our aim is obtained. Otherwise, is oscillatory about .Let
In the following, we will show that for being large enough. There exist two positive constants such that
Since is continuous and bounded and is not effected by impulses, we conclude that is uniformly continuous. Hence there exists a constant (with and is independent of the choice of ) such that for all .If , our aim is obtained.If , from the second equation of (1.3), we have that for . Then we have for since . It is clear that for .If , then we have that for . Next, we will show that for . In fact, if not, there exists a such that for and . When is large enough, from (3.37) and the first equation of (1.3), we have
Similarly, we know that there exists , for that
Then the inequality holds true for . On the other hand, we have from the second equation of (1.3) that
This is a contradiction to . Therefore, for .Since this kind of interval is arbitrarily chosen, we get that for being large enough. In view of our arguments previously, the choice of is independent of the positive solution of (1.3) which satisfies that for sufficiently large . This completes the proof.

Theorem 3.4. If , the system (1.3) is permanent; that is, there exist two positive constants , such that for being large enough.

Proof. Suppose that is any positive solution of system (1.3). From the first and third equations of (1.3), we have that
Similarly, we can get such large enough and small enough that
Set
Then is a bounded compact region which has positive distance from coordinate axes. By Lemma 3.3, one obtains that every solution of system (1.3) eventually enters and remains in the region . The proof of Theorem 3.4 is completed.

4. Numerical Simulation and Conclusion

To verify the theoretical results obtained in this paper, we will give some numerical simulations.

Under the continuous control strategy, we consider the hypothetical set of parameter values as , with . Through calculation, we know and .(i)If , then according to Theorem 2.1, we know the disease-free equilibrium of system (1.2) is local stable for this case (see Figures 1, 2, and 3). (ii)If , through calculation, we know . Then according to Theorem 2.2, the positive equilibrium of system (1.2) is local stable for this case (see Figures 4, 5, and 6).

Figure 1: Time series of with different initial values and parameters .

Figure 2: Time series of with different initial values and parameters .

Figure 3: Phase diagram of with different initial values and parameters .

Figure 4: Time series of with different initial values and parameters .

Figure 5: Time series of with different initial values and parameters .

Figure 6: Phase diagram of with different initial values and parameters .

Its epidemiological implication is that if time delay is greater than some key value , then diseased plants will disappear in local scope. In contrast, if time delay is less than some key value , then susceptible plants and diseased plants will coexist in local scope.

Under the impulsive control strategy, We consider the hypothetical set of parameter values as with .(i)We consider the susceptible plants rate . Through calculation, we have . Then according to Theorem 3.4, we know that system (1.3) is permanence, for this case (see Figures 10, 11, and 12).(ii) If we decrease the susceptible plants rate to 0.3, through calculation, we know . Then according to Theorem 3.1, the disease-free periodic solution of system is globally attractive, for this case (see Figures 7, 8, and 9).

Figure 7: Time series of with parameters , .

Figure 8: Time series of with parameters , .

Figure 9: Phase diagram of and with parameters , .

Figure 10: Time series of with parameters , .

Figure 11: Time series of with parameters , .

Figure 12: Phase diagram of and with parameters , .

We have other hypothetical parameter values under the impulsive control strategy as . with (i)We consider the susceptible plants rate . Through calculation, we have . Then according to Theorem 3.4, we know that system (1.3) is permanence, for this case (see Figures 16, 17, and 18 ).(ii) If we decrease the susceptible plants rate to 0.4, through calculation, we know . Then according to Theorem 3.1, the disease-free periodic solution of system is globally attractive, for this case (see Figures 13, 14, and 15).

Figure 13: Time series of with parameters , .

Figure 14: Time series of with parameters , .

Figure 15: Phase diagram of and with parameters , .

Figure 16: Time series of with parameters , .

Figure 17: Time series of with parameters , .

Figure 18: Phase diagram of and with parameters , .

Its epidemiological implication is that we took such a strategy by improving planting susceptible plants in practice; as a result, if the susceptible plants rate is greater than some key value , both susceptible plants and diseased plants will coexist. In contrast, if we decrease the susceptible plants rate and make it less than some key value , then diseased plants will die out at length. In a word, we find that the impulse plants rate has played a very important role in the actual plant epidemic prevention.

In this paper, delay SIS plant epidemic model is constructed and investigated. We proposed two different control strategies in the model. Our primary results are to compare the difference between the two control methods. Firstly, we consider continuous cultural control strategy by continuous replanting of healthy plants. We come to the conclusion that if , then diseased plants will disappear in local scope where . And if , then diseased plants will exist for a long time in local scope. Secondly, impulsive control strategy of plant disease model is considered; in this case, we get that if , then diseased plants will disappear finally where . And if , then diseased plants will exist for a long time where . We think that our results will offer help to the actual plant infectious disease management.